GRADEX-DUAL: A Dual Framework Combining Point and Regional Frequency Analysis for GRADEX Parameter Estimation in Basins with Sparse Rain Gauge Networks

Abstract

GRADEX-DUAL v1.0, an open-source computational framework implemented in R, is presented. It combines two complementary methodological layers for estimating the GRADEX parameter —defined as the slope of the annual maximum precipitation quantile function on Gumbel probability paper between return periods T = 10 and T = 100 years—, which is fundamental for the GRADEX-IDF methodology applied to hydrological design with HEC-HMS.

The first layer follows the classic pointwise approach of Guillot and Duband (1967): fitting four probability distributions per station (Gumbel, Log-Normal, Pearson III, and Gamma) using L-moments and maximum likelihood; two-stage model selection (Anderson-

Darling followed by minimum RMSE); and regionalization to the gauging point using Inverse Distance Weighting (IDW) with leave-one-out (LOO) cross-validated optimal exponent or Ordinary Kriging with automatic variogram fitting. A multi-metric scoring rule (RMSE + R² + |BIAS|) selects the better spatial method, with an anti-degenerate-Kriging safeguard that disqualifies Kriging whenever its LOO prediction-to-observation variance ratio falls below 0.10 —indicating variogram collapse to a near-constant predictor.

The second layer implements the Hosking and Wallis (1997) regional frequency analysis: record-length-weighted regional L-moments, selection of the optimal regional distribution among five candidate distributions (GEV, GLO, GNO, PE3, and GPA) using the Monte Carlo-calibrated goodness-of-fit Z statistic, and 95% confidence intervals computed using the H&W bootstrap simulation algorithm.

A three-gate quality-control weighting system integrates both estimates: the H&W weight is nullified when the bootstrap confidence interval amplitude exceeds 100%, the spatial weight is nullified when the LOO cross-validation R² falls below 0.3, and Kriging is disqualified when its LOO variance ratio falls below 0.10. Prior diagnostics include the Ljung-Box test for serial independence, the Hosking-Wallis H statistic for regional homogeneity, the discordancy measure D, Mann-Kendall trend tests with Theil-Sen slope estimator, a non-stationarity sensitivity analysis on detrended series, and a Monte Carlo validation experiment against synthetic regions with known true GRADEX.

The framework is evaluated on a nine-station IDEAM network in the upper Porce River basin (Antioquia, Colombia, 254 station-years). The Porce is a major tributary of the Nechí River, which in turn flows into the Cauca River and ultimately into the Magdalena, so the study area belongs hydrographically to the lower Cauca macro-basin.

Results show that even with a homogeneous region (H = −0.92) and a network surpassing the H&W minimum threshold, the bootstrap confidence interval amplitude reaches 121% and the spatial R² collapses to 0.006 once the degenerate-Kriging safeguard is activated, triggering the framework’s fallback regime with a ±30% sensitivity range. A Monte Carlo experiment (200 replicates of 9-station, 30-year synthetic regions sampled from a GEV with κ = −0.10) confirms an empirical coverage of 85.0% within the ±30% interval, and quantifies a systematic underestimation bias of −20.7% attributable to fitting Gumbel L-moments to data drawn from a heavy-tailed GEV.